Power-Partible Reduction and Congruences for Ap\'ery Numbers

TL;DR

This paper develops a new reduction method for holonomic sequences and uses it to establish novel congruences for Apéry numbers, revealing deeper number-theoretic properties of these special sequences.

Contribution

Introduction of power-partible reduction for P-recursive sequences and derivation of new congruences for Apéry numbers.

Findings

Proved congruences for sums involving Apéry numbers modulo p^3.

Established existence of specific constants for these congruences.

Extended understanding of number-theoretic properties of Apéry numbers.

Abstract

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Ap\'ery numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there exists an integer $\tilde{c}_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv \tilde{c}_r p \pmod {p^3} \end{equation*} holds for any prime $p>3$.